Theorem spw 2143

Description: Weak version of the specialization scheme sp 2226. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2226 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2226 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2188 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2226 are spfw 2142 (minimal distinct variable requirements), spnfw 2106 (when 𝑥 is not free in ¬ 𝜑), spvw 2088 (when 𝑥 does not appear in 𝜑), sptruw 1907 (when 𝜑 is true), and spfalw 2107 (when 𝜑 is false). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)

Hypothesis

Ref	Expression
spw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
spw	⊢ (∀𝑥𝜑 → 𝜑)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem spw

Step	Hyp	Ref	Expression
1		ax-5 2011	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
2		ax-5 2011	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
3		ax-5 2011	. 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
4		spw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	1, 2, 3, 4	spfw 2142	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198  ∀wal 1656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1881

This theorem is referenced by:  hba1w  2151  spaev  2154  ax12w  2186  bj-ssblem1  33168  bj-ax12w  33201